Tensor LEED analyses have been made for the Rh(111)-(root 3 x root 3)R30 degrees-S and Rh(lll)-c(4 x 2)-S surface structures formed by S chemisorbed at 1/3 and 1/2 monolayer coverages respectively. For the lower-coverage form, S adsorbs on the regular three-coordinate sites which continue the fee packing sequence; the S-Rh bond lengths are indicated to equal 2.23 Angstrom, and relaxations in the metallic structure are negligible. In the c(4 x 2) form, the adsorption occurs equally on both types of three-coordinate site (fee and hcp), although some surface Rh atoms bond to two S atoms while others bond to just one, and this sets up some interesting relaxations. Specifically, the S atoms displace laterally from the centre of the three-fold sites by 0.20 to 0.29 Angstrom, and the first metal layer is buckled by about 0.23 Angstrom. The first-to-second interlayer spacing in the metal expands to 2.26 Angstrom from the bulk value of 2.20 Angstrom. The average S-Rh bond lengths equal 2.22 Angstrom, and so they are not significantly changed from that in the low-coverage form. The structural evolution for S chemisorbed on the (111) surface of rhodium with increasing coverage is compared with the corresponding evolution on the Rh(110) surface.